#220532

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(9,10,16,13)(11,20,23,33)(12,24)(14,29,15,19)(17,30,28,22)(18,31)(21,36)(25,26)(27,34)(32,39)(35,37)(38,40);(9,11,16,23)(10,17,13,28)(12,21,27,39)(14,30,15,22)(18,32,25,36)(19,33,29,20)(24,35,34,40)(26,38,31,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,21)(13,25)(14,26)(15,31)(16,27)(17,32)(19,34)(20,35)(22,37)(23,39)(24,29)(28,36)(30,38)(33,40);(9,13,16,10)(11,20,23,33)(12,25,27,18)(14,19,15,29)(17,30,28,22)(21,35,39,40)(24,26,34,31)(32,38,36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127728);